STACKED BASES FOR HOMOGENEOUS COMPLETELY DECOMPOSABLE GROUPS

Generalizing a theorem by P. Hill and C. Megibben, fixing a rational group R, we characterize by numerical invariants R-presentations of a group G, namely, short exact sequences of the form 0 → A → X → G → 0, where A and X are homogeneous completely decomposable groups of the same type R. This characterization sets afloat the class of the “uniquely R-presented groups”. This class is investigated in connection with the extension to arbitrary groups of the Warfield equivalence between categories of torsionfree abelian groups induced by the functors Hom(R, –) and R ? ?. As an application, the stacked bases theorem proved by J. Cohen and H. Gluck in 1970 is extended to arbitrary pairs of homogeneous completely decomposable abelian groups of the same type.

     

Representations of finite partially ordered sets over commutative artinian uniserial rings

Categories of representations of finite partially ordered sets over commutative artinian uniserial rings arise naturally from categories of lattices over orders and abelian groups. By a series of functorial reductions and a combinatorial analysis, the representation type of a category of representations of a finite partially ordered set S over a commutative artinian uniserial ring R is characterized in terms of S and the index of nilpotency of the Jacobson radical of R. These reductions induce isomorphisms of Auslander-Reiten quivers and preserve and reflect Auslander-Reiten sequences. Included, as an application, is the completion of a partial characterization of representation type of a category of representations arising from pairs of finite rank completely decomposable abelian groups.     

Factoring by subsets of cardinality of prime power

Rédei's theorem asserts that if a finite abelian group is expressed as a direct product of subsets of prime cardinality, then at least one of the factors must be periodic. (A periodic subset is a direct product of some subset and a nontrivial subgroup.) A. D. Sands proved that if a finite cyclic group is the direct product of subsets each of which has cardinality that is a power of a prime, then at least one of the factors is periodic. We prove that the same conclusion holds if a general finite abelian group is factored as a direct product of cyclic subsets of prime cardinalities and general subsets of cardinalities that are powers of primes provided that the components of the group corresponding to these latter primes are cyclic.     

On pure subgroups of locally compact abelian groups

In this note, we construct an example of a locally compact abelian group G = C × D (where C is a compact group and D is a discrete group) and a closed pure subgroup of G having nonpure annihilator in the Pontrjagin dual $\hat{G}$, answering a question raised by Hartman and Hulanicki. A simple proof of the following result is given: Suppose ${\frak K}$ is a class of locally compact abelian groups such that $G \in {\frak K}$ implies that $\hat{G} \in {\frak K}$ and nG is closed in G for each positive integer n. If H is a closed subgroup of a group $G \in {\frak K}$, then H is topologically pure in G exactly if the annihilator of H is topologically pure in $\hat{G}$. This result extends a theorem of Hartman and Hulanicki.Received: 4 April 2002     

Doubly transitive permutation groups with abelian stabilizers

Summary We prove that any doubly transitive permutation group with abelian stabilizers is the group of linear functions over a suitable field. The result is not new: for finite groups it is well known, for infinite groups it follows from a more general theorem of W. Kerby and H. Wefelscheid on sharply doubly transitive groups in which the stabilizers have finite commutator subgroups. We give a direct and elementary proof.     

Tight subgroups in torsion-free Abelian groups

LetX be a torsion-free abelian group. We study the class of all completely decomposable subgroups ofX which are maximal with respect to inclusion. These groups are called tight subgroups ofX and we state sufficient conditions on a subgroup to be tight. In particular we consider tight subgroups of bounded completely decomposable groups. For those we show that every regulating subgroup is tight and we characterize the tight subgroups of finite index in almost completely decomposable groups. The second author was supported by a MINERVA fellowship.     

Splitting Mixed Groups of Finite Torsion-Free Rank

Abstract

First, we give a necessary and sufficient condition for torsion-free finite rank subgroups of arbitrary abelian groups to be purifiable. An abelian group G is said to be a strongly ADE decomposable group if there exists a purifiable T(G)-high subgroup of G. We use a previous result to characterize ADE decomposable groups of finite torsion-free rank. Finally, in an extreme case of strongly ADE decomposable groups, we give a necessary and sufficient condition for abelian groups of finite torsion-free rank to be splitting.     

Locally compact abelian groups with symplectic self-duality

Is every locally compact abelian group which admits a symplectic self-duality isomorphic to the product of a locally compact abelian group and its Pontryagin dual? Several sufficient conditions, covering all the typical applications are found. Counterexamples are produced by studying a seemingly unrelated question about the structure of maximal isotropic subgroups of finite abelian groups with symplectic self-duality (where the original question always has an affirmative answer).     

Representations of Posets and Indecomposable Torsion-Free Abelian Groups

Representations of posets in certain modules are used to find indecomposable almost completely decomposable torsion-free abelian groups. For a special class of almost completely decomposable groups we determine the possible ranks of indecomposable groups and show that the possible ranks are realized by indecomposable groups in the class.     

Characterization of completely decomposable quotient divisible abelian groups by their endomorphism semigroups

We study abelian groups in the class of completely decomposable quotient divisible abelian groups which are determined in this class by their endomorphism semigroups.     

On Torsion-Free Minimal Abelian Groups

ABSTRACT

An abelian group is said to be minimal if it is isomorphic to all its subgroups of finite index. In this article we show that torsion-free groups which are complete in their ?-adic topology or are of p-rank not greater than 1, for all primes p, are minimal. A criterion is found for the minimality of all finite rank and for large classes of infinite rank completely decomposable groups. Separable minimal groups are also considered.     

Some Nasty reflexive groups

In {\it Almost Free Modules, Set-theoretic Methods}, Eklof and Mekler [5,p. 455, Problem 12] raised the question about the existence of dual abelian groups G which are not isomorphic to . Recall that G is a dual group if for some group D with . The existence of such groups is not obvious because dual groups are subgroups of cartesian products and therefore have very many homomorphisms into . If is such a homomorphism arising from a projection of the cartesian product, then . In all `classical cases' of groups {\it D} of infinite rank it turns out that . Is this always the case? Also note that reflexive groups G in the sense of H. Bass are dual groups because by definition the evaluation map is an isomorphism, hence G is the dual of . Assuming the diamond axiom for we will construct a reflexive torsion-free abelian group of cardinality which is not isomorphic to . The result is formulated for modules over countable principal ideal domains which are not field. Received July 1, 1999; in final form January 26, 2000 / Published online April 12, 2001     

On a class of butler groups

A complete classification, both up to quasi isomorphism and up to isomorphism, is given of the strongly indecomposable torsionfree abelian groups that occur as quotients of a finite rank completely decomposable torsionfree group modulo a rank 1 subgroup. This research was done while the second author held a visiting professorship at Tulane University.     

Transfer and Tate’s theorem

Let H be a subgroup of a finite group G, and assume that p is a prime that does not divide |G : H|. In favorable circumstances, one can use transfer theory to deduce that the largest abelian p-groups that occur as factor groups of G and of H are isomorphic. When this happens, Tate’s theorem guarantees that the largest not-necessarily-abelian p-groups that occur as factor groups of G and H are isomorphic. Known proofs of Tate’s theorem involve cohomology or character theory, but in this paper, a new elementary proof is given. It is also shown that the largest abelian p-factor group of G is always isomorphic to a direct factor of the largest abelian p-factor group of H. Received: 17 June 2008     

Almost maximally almost-periodic group topologies determined by T-sequences

A sequence {an} in a group G is a T-sequence if there is a Hausdorff group topology τ on G such that . In this paper, we provide several sufficient conditions for a sequence in an abelian group to be a T-sequence, and investigate special sequences in the Prüfer groups Z(p^∞). We show that for p≠2, there is a Hausdorff group topology τ on Z(p^∞) that is determined by a T-sequence, which is close to being maximally almost-periodic—in other words, the von Neumann radical n(Z(p^∞),τ) is a non-trivial finite subgroup. In particular, n(n(Z(p^∞),τ))?n(Z(p^∞),τ). We also prove that the direct sum of any infinite family of finite abelian groups admits a group topology determined by a T-sequence with non-trivial finite von Neumann radical.     

Prebasic submodules over valuation rings

Summary Basic concepts in the theory of modules over valuation rings are introduced. The notion of height is used to define indicators of elements, whose irregularities are investigated. The indicator leads to the new notion of smoothness, a property which does not originate from abelian groups. Invariants generalizing the finite Ulm-Kaplansky invariants of abelian p-groups, as well as the Baer invariants for completely decomposable torsion-free abelian groups, are defined, and several results relating these invariants of a module to those of submodules are proved. All these concepts lead to the notion of prebasic submodules, which seems to be the right analogue of the basic subgroups in abelian groups.The authors gratefully acknowledge the support of the National Science Foundation and the Italian C.N.R.     

Idempotent reducts of abelian groups

In this paper we construct two maps between the polynomials of abelian groups and the polynomials of idempotent reducts of abelian groups and show that these can be used to “lift” the finite basis property from abelian groups to their idenpotent reducts. It follows that this equational class of all idempotent reducts of abelian groups has the finite basis property. The research of this author was supported by a grant from the National Research Council of Canada. Presented by J. Mycielski     

Realizable posets

Certain p-local orders in n-dimensional division algebras over the rational numbers occur as endomorphism rings of torsion-free abelian groups of rank n if and only if an associated finite poset P has a strict faithful representation of dimension less than |P| over the field with p elements. In this note we obtain a simple characterization of those finite posets which do not admit such a representation.Research supported in part by NSF grant DMS-8802833.